Problem: You have found the following ages (in years) of all 4 turtles at your local zoo: $ 51,\enspace 58,\enspace 62,\enspace 27$ What is the average age of the turtles at your zoo? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we have data for all 4 turtles at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{51 + 58 + 62 + 27}{{4}} = {49.5\text{ years old}} $ Find the squared deviations from the mean for each turtle. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $51$ years $1.5$ years $2.25$ years $^2$ $58$ years $8.5$ years $72.25$ years $^2$ $62$ years $12.5$ years $156.25$ years $^2$ $27$ years $-22.5$ years $506.25$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{2.25} + {72.25} + {156.25} + {506.25}} {{4}} $ $ {\sigma^2} = \dfrac{{737}}{{4}} = {184.25\text{ years}^2} $ The average turtle at the zoo is 49.5 years old. The population variance is 184.25 years $^2$.